There are now a number of options for capturing tweets from the Twitter API. All options require a Twitter account and some will require creating an app with Twitter (this requires associating a phone number with your Twitter account) in order to generate the necessary credentials (keys and access tokens).
When using the Twitter API, keep in mind that you are working with tweets that are no more than 7 days old. If you need historical tweets, you most likely would need to purchase them from Twitter or subscribe to a resource that has access to historical Twitter data. The costs of working with historical Twitter data can be high, though there are some collections of Twitter data that have been made publicly accessible (see the "Twitter Data" page).
If you would like to collect your own Twitter data, see the list of software links on the "Collecting Tweets" page.